A characterization of the concept of duality

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In this paper we discuss the abstract concept of duality. We consider various classes: convex functions, s-concave functions, convex bodies and log-concave functions. We demonstrate how very little a-priori information is needed in order to arrive at the concrete formulas for duality in each of the classes. Some of these transforms, such as the Legendre transform or polarity in convexity, are classical and well-know, and some more recently discovered and used. In most cases the information we require is only, essentially, that inequalities between functions are reversed. In another case the information is that the transform exchanges summation with the operation of inf-convolution.

Original languageEnglish
Pages (from-to)42-59
Number of pages18
JournalElectronic Research Announcements of the American Mathematical Society
StatePublished - 2007


  • Convexity
  • Duality
  • Inf-convolution
  • Legendre


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